CALCULUS III EXAM IV QUESTIONS

CALCULUS III EXAM IV PRACTICE EXAM Answers

1. Find the volume of the solid bounded by the graphs of the given equations.

z = xy, z = 0, y = x, x = 1, y = 0 (first octant)

2. Evaluate the double integral by changing to polar coordinates.

3. Find the mass and center of mass of the lamina bounded by the graphs of the given equations with a density equal to kx.

x = 16 - y², x = 0

4. Find the area of the surface given by z = f(x,y) over the region R.

R: The triangle with vertices (0,0), (2,0), and (0,2)

f(x,y) = 2x + 2y

5. Find the mass and the z-coordinate of the center of mass of the solid of density kx bounded by the graphs of the given equations.

z = 4 - x, z = 0, y = 0, y = 4, x = 0

6. Convert the integral from rectangular coordinates to cylindrical coordinates and evaluate the integral.

7. Find the volume of the solid bounded by the graphs of the given equations (given in cylindrical coordinates).

DPGraph Picture with a = 4 DPGraph Picture of the Surfaces with a = 4 Maple Worksheet Picture

8. Find the mass of the sphere x² + y² + z² = a² where the density at any point is two times the distance from the point to the origin.

Here is a DPGraph Picture of a cross section of the sphere with the colorization relating to the density function and a = 4.

9. Use a change of variables (and the Jacobian) to find the volume of the solid region lying below the surface z = f(x,y) and above the plane region R. Picture of the solid

f(x,y) = (x+y)e^x-y

R: the region of the xy-coordinate plane bounded by the square with vertices (4,0), (6,2), (4,4), and (2,2)

DPGraph Picture with one unit = 100 on the z-axis

10. You are given a sphere of radius 2. A hole of radius b (b < 2) is drilled through the center of the sphere, the sphere's center being in the middle of the hole. Find b such that the volume of what is left of the sphere equals one half the original volume of the sphere.

Here are five DPGraph pictures relating to problem number 10. The first one is an animation as the hole gets larger and larger. The second one is a semi-transparent animation showing the sphere and a cylinder representing the hole. The third one shows the hole from a different view. In the fourth one you can use the scrollbar to vary the value of b. The fifth one is like the forth one except the opening view is from the front and is a nice one to use the animate menu on (continuously rotate).

DPGraph1 DPGraph2 DPGraph3 DPGraph4 DPGraph5

BONUS--Find the volume of the torus pictured on this Maple Worksheet. This does not require the evaluation of a double integral.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu